1. **Problem Statement:** We want to find the ratio of the initial populations of rats ($R_0$) and skunks ($S_0$) that allows both species to survive in the long term. 2. **Background:** In population dynamics, coexistence often requires that the growth rates and interaction terms balance so neither species goes extinct. 3. **Assuming a Lotka-Volterra type model:** The populations evolve as $$\frac{dR}{dt} = R(\alpha - \beta S)$$ $$\frac{dS}{dt} = S(-\gamma + \delta R)$$ where $\alpha, \beta, \gamma, \delta$ are positive constants representing growth and interaction rates. 4. **Equilibrium condition:** For both species to survive long term, their populations must be stable, so $$\frac{dR}{dt} = 0 \quad \Rightarrow \quad \alpha - \beta S = 0 \Rightarrow S = \frac{\alpha}{\beta}$$ $$\frac{dS}{dt} = 0 \quad \Rightarrow \quad -\gamma + \delta R = 0 \Rightarrow R = \frac{\gamma}{\delta}$$ 5. **Initial ratio for coexistence:** The initial populations must be proportional to these equilibrium values: $$\frac{R_0}{S_0} = \frac{\gamma / \delta}{\alpha / \beta} = \frac{\gamma \beta}{\alpha \delta}$$ 6. **Interpretation:** This ratio ensures that neither species outcompetes the other initially, allowing stable coexistence. **Final answer:** $$\boxed{\frac{R_0}{S_0} = \frac{\gamma \beta}{\alpha \delta}}$$